Convergence of Laplacian Eigenmaps and its Rate for Submanifolds with Singularities
This provides a theoretical foundation for spectral methods in machine learning on complex data structures, though it appears incremental as it extends prior work to singularities.
The paper tackles the problem of approximating the Laplacian on submanifolds with singularities using random point graphs, achieving a convergence rate of O((log n/n)^(1/(m+2))) for eigenvalues.
In this paper, we give a spectral approximation result for the Laplacian on submanifolds of Euclidean spaces with singularities by the $ε$-neighborhood graph constructed from random points on the submanifold. Our convergence rate for the eigenvalue of the Laplacian is $O\left(\left(\log n/n\right)^{1/(m+2)}\right)$, where $m$ and $n$ denote the dimension of the manifold and the sample size, respectively.